Our considerations thus far have been independent of translational symmetries.
In this section, we extend our arguments to lattice models with discrete translation symmetries.
As discussed in Sec. , disclinations in lattices are classified into topological equivalence classes according to their rotation and translation holonomy.
Real-space representations of topological crystalline phases naturally including translation symmetries can be constructed using the framework of topological crystals [1]}.
Below, we briefly review and discuss the essential steps of the topological-crystal construction applied to lattices with rotation symmetries.
Moreover, we extend the recipe developed in Ref. [1]} by showing how to relate the constructed real-space representation to weak and higher-order topological phases obtained from other classification schemes [3]}, [4]}, [5]}.
We then apply the topological-crystal construction to determine the existence of anomalous states at disclinations of all types.
As the topological crystal construction was originally performed for periodic samples, we briefly comment in App. REF  on the validity of the approach for finite size samples or in the presence of inhomogeneities. Finally, we provide a summary of the main results of this section.
